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The book starts with a thorough introduction to connections and holonomy groups, and to Riemannian, complex and Kahler geometry. Then the Calabi conjecture is proved and used to deduce the existence of compact manifolds with holonomy SU(m) (Calabi-Yau manifolds) and Sp(m) (hyperkahler manifolds). These are constructed and studied using complex algebraic geometry. The second half of the book is devoted to constructions of compact 7- and 8-manifolds with the exceptional holonomy groups 92 and Spin(7). Many new examples are given, and their Betti numbers calculated. The first known examples of...

This graduate level text covers an exciting and active area of research at the crossroads of several different fields in Mathematics and Physics. In Mathematics it involves Differential Geometry, Complex Algebraic Geometry, Symplectic Geometry, and in Physics String Theory and Mirror Symmetry. Drawing extensively on the author's previous work, the text explains the advanced mathematics involved simply and clearly to both mathematicians and physicists. Starting with the basic geometry of connections, curvature, complex and Kahler structures suitable for beginning graduate students, the text...

This graduate level text covers an exciting and active area of research at the crossroads of several different fields in mathematics and physics. In mathematics it involves Differential Geometry, Complex Algebraic Geometry, Symplectic Geometry, and in physics String Theory and Mirror Symmetry. Drawing extensively on the author's previous work, the text explains the advanced mathematics involved simply and clearly to both mathematicians and physicists. Starting with the basic geometry of connections, curvature, complex and Kahler structures suitable for beginning graduate students, the text...

If $X$ is a manifold then the $mathbb R$-algebra $C^infty (X)$ of smooth functions $c:X ightarrow mathbb R$ is a $C^infty $-ring. That is, for each smooth function $f:mathbb R^n ightarrow mathbb R$ there is an $n$-fold operation $Phi _f:C^infty (X)^n ightarrow C^infty (X)$ acting by $Phi _f:(c_1,ldots ,c_n)mapsto f(c_1,ldots ,c_n)$, and these operations $Phi _f$ satisfy many natural identities. Thus, $C^infty (X)$ actually has a far richer structure than the obvious $mathbb R$-algebra structure. The author explains the foundations of a version of algebraic geometry in which rings or...